Why Our Stringed Instruments Sound Out of Tune
and A Way to Help Correct the Problem

Some months ago, I promised Phil Mann that I would write this little
article because this is such a misunderstood topic and most explanations
are based on folklore, not fact.

Several years ago, I was given the rather undelightful task of having to
tune accordion reeds. This was made somewhat easier than one might think
by the fact that my late father had been extremely aware of the problems
involved, and he had provided me with a Conn Strobotuner. This was long
before the advent of the inexpensive electronic tuner so easily
obtainable today. I was quite confused about why certain instruments
sounded in tune with the accordion and others did not. Later, I
undertook a study of tuning and learned several things from the
experience.

To understand tuning, you must understand temperment, and to understand
temperment, you must understand "Just" intonation. (This is pronounced
like the first syllable in "justice," not as a rhyme with "fist.") Just
intonation is a system of tuning that is based on the normal series of
harmonics that are generated by a vibrating body. In the case of a
guitar or a banjo, this vibrating body is the string. These harmonics
are expressed in terms of ratios of small numbers. They are as follows:

Whenever you pluck a string, it will vibrate in all of these modes at
once. Normally, the fundamental will be the loudest of the tones, and
each of the higher harmonics will be increasingly less strong. It is the
amount of each of these harmonics that gives an instrument its unique
timbre or tonal quality.

When one of these "just" intervals is played against the fundamental,
there is no "beat" between the fundamental and that interval. Just
intervals sound "clean." Many choral experts theorize that a capella
singers use just intonation.

We tend to hear in terms of just intonation.

In instruments that have only 7 notes per octave, i.e. diatonic
instruments, such as the harmonica, bagpipe, Celtic harp, lyre, etc.
there is no problem tuning the instrument in such a way that all the
intervals are just. However, when instruments are chromatic, such as the
piano, the guitar or the banjo, things soon go awry.

The traditional way to tune a keyboard instrument starts with just
intervals. A particular note, such as Middle C is played on a reference
instrument, such as a tuning fork. Then the fifth below the C is tuned
so that it does not beat against the C, giving us a just F. Next the
octave above the F is tuned, and then the Bb below that. Next, the G
above the C is tuned, then the G an octave below that. This process
continues with the D, the A and the E. The tuning goes "down a fifth, up
an octave" and repeats.

When we start getting to the B, itself and then the remaining
accidentals, F#, Eb, C# and Ab, We start getting into trouble. For some
unknown reason, the old keyboard tuners of long ago decided that B was
going to be the note that would get the "leftovers."

"Wait a minute," you ask. "What leftovers?" Remember. I said that things
were pretty simple until you started going chromatic? Here's why. If you
try to set up a chromatic scale based on A=440, but you do all of your
divisions by a 3/2 ratio, B ends up with a frequency of 247.5 Hz. If you
are figuring your ratios from C=261.6255 (normal middle C from an
electronic tuner) B becomes 248.3398 Hz. However, if you are tuning
from an electronic tuner, B is 246.941 Hz. This is somewhat confusing.
How can a note have 3 different pitches?

This is due to a practice called "temperment." In order to create a
chromatic scale that is the same on all instruments (basically), since
1885 or so, piano tuners have used a scale that is based on a
mathematically determined octave. In this scale, an octave equals 1200
cents. A semitone, that is, the distance between two adjacent notes on
the piano, or one fret on a fretted instrument, is equal to 100 cents.
The interval of a fifth is 700 cents. However, a just fifth is 702
cents. When you add up all of the 2 cent leftovers that gives you an
error of 24 cents. The idea of temperment is to spread these 24 cents
out equally through the octave.

An electronic tuner, contrary to what most musicians believe, does not
put you in tune. It puts you in temperment. Temperment is the amount of
"out of tune-ness" that is necessary so that we can play in all keys,
and sound equally in tune (or equally out of tune, if you will!) The
octave and the unison are the only "just" intervals left in our equal
tempered tuning system.

Over the centuries, there have been at least 64 different schemes to
accomplish this, each with differing results. The piano tuner resolves
this by tuning just, then checking a set of key intervals to make sure
that all of the major thirds are out by an equal amount, etc.

In other words, the problem is that, to a great degree, we hear in terms
of just intonation, but we play instruments that use equal temperment.
We hear in tune and play out of tune.

The Guitar and Banjo Tuning Method.

To tune your stringed instrument correctly, there are some things you
can do to make these adjustments.

First, place your bridge as accurately as possible. If the bridge is
placed so the first string is very slightly sharp at the 12th fret, the
upper octaves will sound more in tune than if it is placed perfectly or
flat. Piano tuners routinely tune octaves starting at the F an octave
and a fourth abouve middle C very slightly sharp--"one long roll sharp."
Also, make sure that your strings are reasonably recent, but properly
set, so that they do not stretch too readily.

For the guitar. Tune your first string E, either with an electronic
tuner or by using an A=440 tuning fork. Fret the first string at the 3rd
fret, and use this G to tune the open G string. Make sure there is no
beat at all between these two notes. Next, tune the fourth string by
fretting it at the second fret, and tuning this note E to the first
string E. Again, there must be no beat. Tune the open low E on the 6th
string to the E on the second fret of the 4th string. Tune the fifth
string open A to the second fret of the third string G. Then tune the
open second string to the B on the second fret of the fifth string. Now
make all of your checks. Test the D at the third fret of the B string
against the open fourth string. There should be no beat. If there is, go
back and check your D string against the E strings, by fretting it at
the second fret. It may take several cycles of tuning to get the
instrument in tune, possibly just to equalize the string tension on the
neck and body. However, on most instruments, this method will work out
perfectly, even on those with incorrectly placed bridges.

You may also start this method by tuning the A string against an A=440
tuning fork. From there, tune the B, then the G, the D and the E
strings.

This method spreads the errors out equally over the fretboard.

For the banjo. Tune the open G string (third string) to an electroninc
tuner or to your guitar player. You may also tune your G string at the
second fret to an A=440 tuning fork. Tune the G at the 5th fret of the
first string against this open 3rd string, Tune the open 5th string to
the open 3rd string. Tune the open 4th string to the open first string.
Tune the 8th fret of the second string to the open 5th string. Then
check the 3rd fret of the second string against the open D strings.
Repeat until the tension on the strings and the neck is equalized. This
is far more critical than on the guitar because of the length of the
neck of the banjo and its flexibility. Note: Because the banjo is so
"lively," you might find it necessary to mute the strings you are not
tuning, just to keep the natural overtones out of the way.

For more information on temperment, there is a massive book by Owen
Jorgensen called "Tuning." It is concerned with temperments of keyboard
instruments and the various schemes used for them. It is a wealth of
information on a poorly understood subject. You can get this from
college book stores or you can find it at some of the better remainder
houses. My copy came from Half Price books and cost me $20.00. It may be
available in your local or college library. There is also lots of
information about temperment in the Grove Dictionary of Music, However,
Jorgensen's research is far more exhaustive and has pointed out some
common errors in the Grove work.